Rearrangeable multiconnection switching networks constructed using combinatorial designs

ABSTRACT

A switching network where a large class of combinatorial designs which are well known in the mathematical literature are for the first time applied to advantageously define the pattern of permanent connections effected between network input channels and initial network crosspoints, illustratively by a connection arrangement of a two-stage, rearrangeable network. The class of combinatorial designs comprises designs of three types: (1) block designs, (2) orthogonal arrays, and (3) difference sets. Each of these is used in a unique manner to derive an advantageous pattern of permanent connections.

CROSS REFERENCE TO RELATED APPLICATION

This is a division of application Ser. No. 06/946,431, filed Dec. 23,1986 now U.S. Pat. No. 4,887,079.

TECHNICAL FIELD

This invention relates to switching networks and, more particularly, tomulticonnection, broadcast switching networks that are rearrangeable toavoid blocking.

BACKGROUND OF THE INVENTION

It is possible for a customer facility connected to a multistageswitching network to occasionally be blocked from being connected asdesired because the network happens to be interconnected in a mannerthat prevents effecting the desired interconnection. This, of course, isan undesirable situation which, in an appropriately designed network, isremedied by dismantling one or more existing interconnections andrearranging the interconnection paths to accommodate the new request.When such a rearrangement is possible, it is said that the newassignment, which is the new set of interconnections desired to beestablished, is realizable. A switching network which can realize allpossible assignments without rearranging existing connections is said tobe non-blocking, while a network which can realize all possibleassignments only by occasionally rearranging existing connections issaid to be rearrangeable. Typical rearrangeable networks have many fewercrosspoints than their non-blocking counterparts. An illustrativerearrangeable network, along with the common control equipmentassociated therewith, is disclosed in U.S. Pat. No. 3,129,407 issued toM. C. Paull on Apr. 14, 1964. Other rearrangeable networks are disclosedin the article by V. E. Benes, "On Rearrangeable Three-Stage ConnectingNetworks," Bell System Technical Journal, Vol. 41, No. 5, September1962, pages 1481-1492 and in U.S. Pat. No. 4,038,638 issued to F. K.Hwang on July 26, 1977. Each of these known switching networks is,however, a rearrangeable point-to-point network rather than arearrangeable multiconnection network. Further each of these networkscomprises three or more stages of switching. In applications wherenetwork distortion and delay parameters directly related to the numberof crosspoints required to effect a given connection are important, thetransmission quality obtainable through such three-stage networks istherefore limited.

U.S. Pat. No. 4,556,007 issued to G. W. Richards on Jan. 21, 1986,discloses a two-stage, multiconnection switching network including aninnovative connection arrangement that permanently connects each networkinput channel to a multiplicity of first stage switch inlets in apredetermined pattern. For any given assignment of input channels to thenetwork output channels connected to a second stage switch, the networkcan always be arranged such that each input channel is connected by adifferent first stage switch to the second stage switch and therefore tothe output channels which are assigned that input channel. Accordingly,the switching network is a rearrangeable multiconnection network thatavoids blocking. The innovative connection pattern advantageouslyeliminates the need for additional stages of switching thereby reducingboth the total number of network crosspoints and the number ofcrosspoints used to effect each interconnection.

The above-mentioned Richards patent discloses a single explicit designmethod for designing the connection arrangement. Although the method iseffective in constructing rearrangeable, two-stage networks, the absenceof other, more general design methods limits the freedom of networkdesigners in optimizing their designs in view of practical networkconstruction parameters such as the maximum number of terminals percircuit pack and the power dissipated per circuit pack, limits on thesize of failure groups and the number of first stage switches each inputchannel is connected to, and constraints on the number of network inputand output channels (e.g., restricting to powers of two).

In view of the foregoing, a recognized problem in the art is the limitedclass of known rearrangeable, two-stage networks.

SUMMARY OF THE INVENTION

The foregoing problem is solved and a technical advance is achieved inaccordance with the principles of the invention in a switching networkwhere a large class of combinatorial designs which are well known in themathematical literature are for the first time applied to advantageouslydefine the pattern of permanent connections effected between networkinput channels and initial network crosspoints, illustratively by aconnection arrangement of a two-stage, rearrangeable network. The classof combinatorial designs comprises designs of three types: (1) blockdesigns, (2) orthogonal arrays, and (3) difference sets. Each of theseis further described herein and is used in a unique manner to derive anadvantageous pattern of permanent connections.

One embodiment of a two-stage, rearrangeable multiconnection switchingnetwork in accordance with the invention connects N₁ input channels ton₂ output channels and comprises v first stage switches and a secondstage switch, e.g., rectangular arrays. Each first stage switch has atmost r inlets and at least one outlet. A single link connects a givenfirst stage switch and the second stage switch. The second stage switchhas n₂ outlets each connected to one of the n₂ output channels. Aconnection arrangement connects each of the first stage switch inlets toan associated predetermined input channel in accordance with a blockdesign having five parameters: b, c, r, M, and w. The (b,c,r,M,w) blockdesign is a matrix having the unique switch designations of the firststage switches as elements, and having the switch designations in agiven row of the block design define the first stage switches connectedto a given input channel associated with the given row.

In another embodiment of the invention, the connection pattern of theconnection arrangement is in accordance with a M×N² connection matrixwhere the elements of row i of the matrix are obtained by adding iN tocorresponding elements of an orthogonal array of order N and depth M.The array has the symbols 0,1, . . . N-1 as elements. The matrix has theunique switch designations d₀,d₁, . . . d_(v-1), as elements and eachcolumn of the matrix is associated with at most w of the input channels.Each element e of a given column of the matrix defines that the inputchannels associated with the given column are connected to the firststage switch having the switch designated d_(e).

In a third embodiment of the invention, the connections effected by theconnection arrangement are in accordance with a Mu×r connection matrixderived from M-1 (u,r,w) difference sets D_(i) ={a_(o) ^(i), . . .a_(r-1) ^(i) }, 1≦i≦M-1. (The superscript i does not denoteexponentiation but rather identifies the elements such as a_(o) ^(i) asbelonging to the difference set D_(i).) The difference sets have threeparameters u, r and w. The connection matrix has Mu rows comprising Msets, s₀, s₁, . . . s_(M-1), each having u rows. The unique channeldesignations of the input channels each occur exactly once in set s₀.For any integer i, 1≦i≦ M-1, and any integer j, 0≦j≦r-1, column j of sets_(i) is obtained by rotating column j of set s₀ by a_(j) ^(i). Thechannel designations occurring in a given row of the matrix define theinput channels connected to a given first stage switch associated withthe given row.

In each of the above-described networks, the following property is true.For any group of n₂ of the input channels, there is a group of n₂ of thefirst stage switches each having one inlet connected to a different oneof that group of n₂ input channels. The connections within that group ofn₂ first stage switches are always rearrangeable to connect a differentone of the group of n₂ input channels to the second stage switch. Theconnections within the second stage switch are therefore alsorearrangeable to connect those input channels to the group of n₂ outputchannels. Accordingly, the network is rearrangeable to avoid theblocking of connections from the group of n₂ input channels to the n₂output channels. In addition, any given input channel is connectible toall of the n₂ output channels. The approach is easily extendible toswitching networks serving any larger number of output channelssignificantly by having multiple links between each first stage switchand the second stage switch. The approach is also extendible by addingsecond stage switches and connecting each additional second stage switchto each first stage switch.

In each of the above-described networks, the connection arrangement andfirst stage switches are replaceable by a partial concentrator havingrows and columns (corresponding to the first stage switch outlets andinlets respectively) and a pattern of crosspoints at certain row-columnintersections defined in an analogous way based on the combinatorialdesigns.

DRAWING DESCRIPTION

FIG. 1 is a block diagram of a two-stage, rearrangeable broadcastnetwork constructed based on a block design shown in. FIG. 2;

FIG. 2 is a block design;

FIG. 3 is a block diagram of an alternative construction of a networkthat is equivalent to the network of FIG. 1;

FIGS. 4 through 6 are block diagrams of networks that are extensions andgeneralizations of the network of FIG. 1;

FIG. 7 is a block diagram of a two-stage, rearrangeable broadcastnetwork constructed based on an orthogonal array shown in FIG. 8;

FIG. 8 is an orthogonal array;

FIG. 9 shows a connection matrix used as an intermediate step in theconstruction of the network of FIG. 7;

FIG. 10 is a block diagram of an alternative construction of a networkthat is equivalent to the network of FIG. 7; FIGS. 11 through 13 areblock diagrams of networks that are extensions and generalizations ofthe network of FIG. 7;

FIG. 14 is a block diagram of a two-stage, rearrangeable broadcastnetwork constructed based on two difference sets shown in FIG. 15;

FIG. 15 is a two difference sets;

FIG. 16 shows a difference table for one of the difference sets of FIG.15;

FIG. 17 shows a connection matrix used as an intermediate step in theconstruction of the network of FIG. 14;

FIG. 18 is a block diagram of an alternative construction of a networkthat is equivalent to the network of FIG. 14;

FIGS. 19 through 21 are block diagrams of networks that are extensionsand generalizations of the network of FIG. 14;

FIGS. 22 through 27 are block diagrams of alternative constructions ofnetworks that are equivalent to the networks of FIGS. 4, 5, 11, 12, 19and 20, respectively;

FIG. 28 is a block diagram of a two-stage rearrangeable broadcastnetwork based on a block design and implemented using a partial; and

FIG. 29, 30, and 31 are block diagrams of additional generalized,two-stage broadcast networks based on block designs and implementedusing partial concentrators.

DETAILED DESCRIPTION Network Constructions BAsed on Block Designs

FIG. 1 is a block diagram of a two-stage, rearrangeable broadcastnetwork 100 constructed based on a combinatorial design referred to as ablock design. FIG. 2 shows the particular block design used to constructnetwork 100. A block design is characterized by five parameters: b, c,r, M and w. In a (b,c,r,M,w) block design, b elements are used toconstruct the rows or blocks, there are a total of c rows each elementappears a total of r times in the c rows, each row has M elements, andevery subset of two elements appears in exactly w rows. For the(6,10,5,3,2) block design of FIG. 2, six elements 0, 1, 2, 3, 4 and 5are used to construct the rows, there are a total of ten rows, eachelement appears a total of five times in the ten rows, each row hasthree elements, and every subset of two elements appears in exactly tworows. Block designs are well known combinatorial structures. Themathematical properties of block designs and the techniques forconstructing such designs are described, for example, in Chapters 10 and15 and Appendix I of the book by Marshall Hall, Jr., CombinatorialTheory, Second Edition (1986), John Wiley & Sons.

Network 100 (FIG. 1) is used to broadcast video signals from ten videovendors, e.g., 151 and 152, to five customer facilities 171 through 175.Network 100 receives video signals in 10 input channels IC0 through IC9and transmits video signals in five output channels OC0 through OC4 eachconnected to one of the customer facilities 171 through 175. Each of thecustomer facilities 171 through 175 includes a receiver, e.g., 171-R,for receiving the output channel video signals and a channel selector,e.g., 171-CS, which transmits connection requests via a communicationpath 181 to a network controller 180 included in network 100.

Network 100 includes six 5×1 first stage switches 110-0 through 110-5,each having five inlets and one outlet, and a single 6×5 second stageswitch 190 having each of six inlets connected to an associated one ofthe first stage switches 110-0 through 110-5 and having each of fiveoutlets connected to one of the output channels OC0 through OC4. The teninput channels are connected to the 30 first stage switch inlets by aconnection arrangement 140. Connection arrangement 140 connects eachfirst stage switch inlet to an associated predetermined one of the inputchannels IC0 through IC9 in accordance with the (6,10,5,3,2) blockdesign of FIG. 2. (Only the connections from input channel IC0 areexplicitly shown in FIG. 1). In the block design, the first stageswitches 110-0 through 110-5 have the unique switch designations 0through 5, respectively. The switch designations in a given row of theblock design define the ones of the first stage switches connected to agiven input channel associated with the given row. For example, row 0defines that switches 110-0, 110-1 and 110-2 having switch designations0, 1 and 2 are connected to input channel IC0. Row 1 defines thatswitches 110-0, 110-1 and 110-5 having switch designations 0, 1 and 5are connected to input channel ICl. Row 2 defines that switches 110-0,110-2 and 110-4 having switch designations 0, 2 and 4 are connected toinput channel IC2, etc. Several observations can be made concerningconnection arrangement 140. First, each of the input channels IC0through IC9 is connected to exactly three first stage switch inlets.Input channel IC0, for example, is connected to inlets of first stageswitches 110-0, 110-1 and 110-2. Second, no pair of first stage switchesintersect in more than two input channels. (A switch is said to beincident to an input channel if the input channel is connected to theswitch. If two switches are incident to the same input channel, they aresaid to intersect in that input channel.) For example, first stageswitches 110-0 and 110-1 intersect in only input channels IC0 and ICl.An important characteristic of connection arrangement 140 can be statedas follows. For any group of five of the input channels IC0 through IC9,there is a group of five of the first stage switches 110-0 through110-5, each having one inlet connected to a different one of that groupof input channels. For example, consider the group of input channelsIC0, IC2, IC5, IC7, and IC8. Each switch of the group of first stageswitches 110-0, 110-2, 110-3, 110-4 and 110-5 has one inlet connected toa different one of that group of input channels. Switch 110-0 has aninlet connected to input channel IC0, switch 110-2 has an inletconnected to input channel IC2, switch 110-3 has an inlet connected toinput channel IC5, switch 110-4 has an inlet connected to input channelIC7 and switch 110-5 has an inlet connected to input channel IC8. It ispossible that for certain sequences of the customer facilities 171through 175 transmitting connection requests for input channels IC0,IC2, IC5, IC7 and IC8, network 100 may temporarily block one or more ofthe requested input channels. However, it is always possible torearrange the connections of the first stage switches such that switches110-0, 110-2, 110-3, 110-4 and 110-5 connect input channels IC0, IC2,IC5, IC7 and IC8, respectively, to inlets of second stage switch 190.The connections within the second stage switch 190 can then berearranged such that the input channels IC0, IC2, IC5, IC7 and IC8 areconnected to the customer facilities 171 through 175 in accordance withthe connection requests. Since this is possible for any group of five ofthe input channels IC0 through IC9, network 100 is a rearrangeablebroadcast network. The arrangement of connections within the six firststage switches 110-0 through 110-5 and the second stage switch 190 iscontrolled by network controller 180 via two communication paths 182 and183.

FIG. 3 is a block diagram of a two-stage, rearrangeable broadcastnetwork 200 that is equivalent to network 100. The 6×5 second stageswitch 290 of network 200 is identical to second stage switch 190 ofnetwork 100. However the connection arrangement 140 and first stageswitches 110-0 through 110-5 of network 100 are replaced with a partialconcentrator 250 in network 200. Partial concentrator 250 comprises tencolumns connected to the input channels IC0 through IC9, and six rowsconnected to the six inlets of second stage switch 290. In a partialconcentrator, a column is said to be incident to a row if there is acrosspoint at the intersection. Similarly, a row is said to be incidentto a column if there is a crosspoint at the intersection. Inconcentrator 250, each of the columns is incident to the rows defined bya corresponding row of the (6,10,5,3,2) block design of FIG. 2. Forexample, the column connected to input channel IC0 is incident to rows0, 1 and 2, the column connected to input channel ICl is incident torows 0, 1 and 5, etc.

FIG. 28 is a block diagram of a two-stage rearrangeable broadcastnetwork 8000. Switching network 8000 (FIG. 28) is used for selectivelyconnecting N₁ input channels to n₂ output channels such that none ofsaid n₂ output channels is connected to more than one of said N₁ inputchannels at any time, N₁ and n₂ being positive integers each greaterthan one, n₂ being at most equal to N₁. Switching network 8000 comprisesa partial concentrator 8010 having N₁ columns each connected to adifferent one of said N₁ input channels, and v rows, said partialconcentrator comprising means for selectively connecting ones of saidcolumns to one of said rows, v being a positive integer at least equalto n₂, and a switch 8020 having n₂ outlets each connected to a differentone of said output channels and v inlets each connected to a differentone of said rows, said switch comprising means for selectivelyconnecting said outlets to said inlets. Partial concentrator 8010 isderivable from a partial concentrator having c columns and b rows andhaving each of said c columns incident to ones of said b rows defined bya corresponding row of a (b,c,r,M,w) block design, where a column issaid to be incident to a row if there is a crosspoint at theintersection, where eahc of said c columns has a unique columndesignation, where the elements of said block design are the columndesignations of said c columns, where b,c,r, and M are positive integerseach greater than one, where w is a positive integer, where M+w isgreater than three, where b is at least equal to v, and where c is atleast equal to N₁.

FIG. 4 is a block diagram of a two-stage rearrangeable broadcast network300 constructed by extending network 100 previously described. Network300 is used to interconnect 10 input channels IC0 through IC9 to 15output channels OC0 through C14. Connection arrangement 340 of network300 is identical to connection arrangement 140 of network 100. Six firststage switches 310-0 through 310-5 are each 5×3 rectangular switches incontrast to the six 5×1 first stage switches 110-0 through 110-5 ofnetwork 100. Network 300 also includes three 6×5 rectangular secondstage switches 390-0, 390-1 and 390-2 each identical to the one secondstage switch 190 of network 100.

FIG. 5 is a block diagram of a two-stage rearrangeable broadcast network400 representing another extension of the network 100 construction.Network 400 is used to interconnect 10 input channels IC0 through IC9 ton₂ output channels OC0 through OC(n₂ -). Connection arrangement 440 ofnetwork 400 is identical to connection arrangement 140 of network 100and connection arrangement 340 of network 300. Six 5×3 first stageswitches 410-0 through 410-5 of network 400 are identical to first stageswitches 310-0 through 310-5 of network 300. However, in network 400,all three outlets of each first stage switch are connected to an 18×n₂rectangular second stage switch 490. By having multiple links from eachfirst stage switch to second stage switch 490, network 400 can servegreater than five output channels and still be rearrangeable to avoidblocking. More generally, each first stage switch could have x links tothe second stage switch.

FIG. 6 is a block diagram of a generalized, two-stage broadcast network500 constructed from a (b,c,r,M,w) block design. Network 500 is used tointerconnect N₁ input channels IC0 through IC(N₁ -1) to n₂ outputchannels OC0 through OC(n₂ -1). Network 500 includes v, r×1 first stageswitches, e.g., 501, 502, a v×n₂ second stage switch 590, and aconnection arrangement 540. Connection arrangement 540 connects each ofthe inlets of the first stage switches to an associated predeterminedone of the input channels in accordance with a (b,c,r,M,w) block design.Each of the first stage switches has a unique switch designation. Theelements of the block design are the switch designations of the firststage switches and the switch designations in a given row of the blockdesign define the ones of the first stage switches connected to a giveninput channel associated with a given row. The construction parametersfor network 500 are summarized in Table 1.

                  TABLE 1                                                         ______________________________________                                        A (b,c,r,M,w) block design is used to                                         construct connection arrangement 540.                                         N.sub.1 > 1, n.sub.2 > 1, n.sub.2 ≦ N.sub.1, v ≧ n.sub.2, r     > 1,                                                                          b ≧ v, c ≧ N.sub.1, M > 1, w > 0, M + w > 3                     CONSTRUCTION PARAMETERS                                                       FOR NETWORK 500                                                               ______________________________________                                    

Note that b is at least equal to v, the number of first stage switches,and c is at least equal to N₁, the number of input channels. Networkscan be derived from network 500 for serving fewer input channels bydeleting rows of the block design and, if possible, eliminating firststage switches. Each of the input channels IC0 through IC(N₁ -1) isconnected to exactly M first stage switch inlets. No pair of first stageswitches intersect in more than w input channels. (This property isreferred to as the w-intersect property.) For any group of n₂ of theinput channels IC0 through IC(N₁ -1), there is a group of n₂ of thefirst stage switches, e.g., 501, 502, each having one inlet connected toa different one of that group of input channels. It has been determinedby mathematical analysis that network 500 is a rearrangeable, broadcastnetwork where n₂ is at most equal to

    M.sup.2 -1)/w+1                                            (1)

where y denotes the smallest integer not less than y. Of course network500 can be extended to serve a greater number, n₂, of output channels byhaving multiple links between each first stage switch and the secondstage switch as in network 400 (FIG. 5). If there are x links betweeneach first stage switch and the second stage switch, the network is arearrangeable, broadcast network where n₂ is at most equal to

    x[(xM+1) (M-1)/w+1].                                       (2)

Network 500 can also be extended by having multiple second stageswitches as in network 300 (FIG. 4).

FIG. 22 is a block diagram of a two-stage, rearrangeable broadcastnetwork 5300 that is equivalent to network 300 (FIG. 4). The 6×5rectangular switches 5390-0, 5390-1, and 5390-2 are identical to thesecond stage switches 390-0, 390-1, and 390-2 of network 300. Howeverthe connection arrangement 340 and first stage switches 310-0 through310-5 of network 300 are replaced with three identical partialconcentrators 5350-0, 5350-1, and 5350-2 in network 5300. Each of thepartial concentrators 5350-0, 5350-1, and 5350-2 is identical to partialconcentrator 250 of network 200 (FIG. 3). Of course the three partialconcentrators 5350-0, 5350-1, and 5350-2 can also be represented as asingle larger partial concentrator. Further the rows of the singlelarger partial concentrator can be reordered arbitrarily for practicaldesign considerations, for example such that identical rows areconsecutive.

FIG. 23 is a block diagram of a two-stage, rearrangeable broadcastnetwork 5400 that is equivalent to network 400 (FIG. 5). The 18×n₂rectangular switch 5490 is identical to second stage switch 490 ofnetwork 400. However the connection arrangement 440 and first stageswitches 410-0 through 410-5 of network 400 are replaced with three (or,more generally, x) identical partial concentrators 5450-0, 5450-1, and5450-2 in network 5400. Each of the partial concentrators 5450-0,5450-1, and 5450-2 is identical to partial concentrator 250 of 30network 200 (FIG. 3).

FIGS. 29, 30, and 31 are block diagrams of additional generalized,two-stage broadcast networks 8100, 8200, and 8300 based on block designsand implemented using partial concentrators. Switching network 8100(FIG. 29) is used for selectively connecting N₁ input channels to n₂output channels such that none of said n₂ output channels is connectedto more than one of said N₁ input channels at any time, N₁ and n₂ beingpositive integers each greater than one, n₂ being at most equal to N₁.Switching network 8100 comprises a plurality of x partial concentrators8110, 8120 each having N₁ columns each connected to a different one ofsaid N₁ input channels, and v rows, said each partial concentratorcomprising means for selectively connecting ones of said columns to onesof said rows, x being a positive integer greater than one and v being apositive integer at least equal to n₂, and a switch 8130 having n₂outlets each connected to a different one of said output channels and xvinlets each connected to a different one of said rows of said x partialconcentrators, said switch comprising means for selectively connectingsaid outlets to said inlets. Each partial concentrator 8110, 8120 isderivable from a partial concentrator having c columns and b rows andhaving each of said c columns incident to ones of said b rows defined bya corresponding row of a,b,c,r,M,w) block design, where a column is saidto be incident to a row if there is a crosspoint at the intersection,where each of said c columns has a unique column designation, where theelements of said block design are the column designations of said ccolumns, where b, c, r and M are positive integers each greater thanone, where w is a positive integer, where M+w is greater than three,where b is at least equal to v, and where c is at least equal to N₁.

Switching network 8200 (FIG. 30) is used for selectively connecting N₁input channels to n₂ output channels such that none of said n₂ outputchannels is connected to more than one of said N₁ input channels at anytime, N₁ and n₂ being positive integers each greater than one, n₂ beingat most equal to N₁. Switching network 8200 comprises a plurality of xpartial concentrators 8210, 8220 each having N₁ columns each connectedto a different one of said N₁ input channels, and v rows, said eachpartial concentrator comprising means for selectively connecting ones ofsaid columns to one of said rows, x being a positive integer greaterthan one and v being a positive integer at least equal to n₂, and aswitch 8230 having n₂ outlets each connected to a different one of saidoutput channels and xv inlets each connected to a different one of saidrows of said x partial concentrators, said switch comprising means forselectively connecting said outlets to said inlets. Each partialconcentrator 8210, 8220 is derivable from a partial concentrator havingc columns and b rows and having each of said c columns incident to onesof said b rows defined by a corresponding row of a (b,c,r,M,1) blockdesign, where a column is said to be incident to a row of there is acrosspoint at the intersection, where each of said c columns has aunique column designation, where the elements of said block design arethe column designations of said c columns, where b,c,r, and M arepositive integers each greater than one, where b is at least equal to v,and where c is at least equal to N₁.

Switching network 8300 (FIG. 31) is used for selectively connecting N₁input channels to N₂ output channels such that none of said N₂ outputchannels is connected to more than one of said N₁ input channels at anytime, N₁ and N₂ being positive integers each greater than one. Switchingnetwork 8300 comprises a plurality of partial concentrators 8310, 8320each having N₁ columns each connected to a different one of said N₁input channels, and v rows, said each partial concentrator comprisingmeans for selectively connecting ones of said columns to ones of saidrows, v being a positive integer at least equal to a positive integer,n₂, that is less than N₂ and at most equal to N₁, and a plurality ofswitches 8330, 8340 each associated with a different one of said partialconcentrators and each having n₂ outlets each connected to a differentone of said output channels and v inlets each connected to a differentone of the v rows of the associated partial concentrator, said eachswitch comprising means for selectively connecting said outlets of saideach switch to said inlets of said each switch. Each partialconcentrator 8310, 8320 is derivable from a partial concentrator havingc columns and b rows and having each of said c columns incident t o onesof said b rows defined by a corresponding row of a (b,c,r,M,w) blockdesign; where a column is said to be incident to a row if there is acrosspoint at the intersection, where each of said c columns has aunique column designation, where the elements of said block design arethe column designations of said c columns, where b, c, r, and M arepositive integers each greater than one, where w is a positive integer,where M+w is greater than three, where b is at least equal to v, andwhere c is at least equal to N₁.

NETWORK CONSTRUCTIONS BASED ON ORTHOGONAL ARRAYS

FIG. 7 is a block diagram of a two-stage, rearrangeable broadcastnetwork 1100 constructed based on a combinatorial design referred to asan orthogonal array. FIG. 8 shows the particular orthogonal array usedto construct network 1100. An orthogonal array of order N and depth M isa matrix with M rows and N² columns in which cells are occupied by Ndistinct symbols and where every pair of rows is orthogonal, i.e., theN² columns produce N² pairs of elements from any two rows, where thoseN² pairs include all N² ordered pairs of the N distinct symbols. Theorthogonal array of FIG. 8 is a matrix of order four and depth threehaving the four distinct symbols 0, 1, 2 and 3. Rows 0 and 2, forexample, are orthogonal because the 16 columns produce the 16 orderedpairs of elements: (0,0), (0,1), (0,2), (0,3), (1,1), (1,0), (1,3),(1,2), (2,2), (2,3), (2,0), (2,1), (3,3), (3,2), (3,1), (3,0).Similarly, rows 0 and 1 are orthogonal and rows 1 and 2 are orthogonal.Orthogonal arrays are well known combinatorial structures. Themathematical properties of orthogonal arrays and the techniques forconstructing such arrays are described, for example, in Chapter 13 ofthe above-referenced book by Hall.

The 3×16 connection matrix shown in FIG. 9 is used as an intermediatestep in the construction of network 1100. The matrix of FIG. 9 isobtained from the orthogonal array of FIG. 8 by adding four to eachelement of row 1 and eight to each element of row 2. In general, iN isadded to each element of row i.

Network 1100 (FIG. 7) is used to broadcast video signals from 32 videovendors, e.g., 1151 and 1152, to six customer facilities 1171 through1176. Network 1100 receives video signals in 32 input channels IC0through IC31 and transmits video signals in six output channels OC0through OC5 each connected to one of the customer facilities 1171through 1176. Each of the customer facilities 1171 through 1176 includesa receiver, e.g., 1171-R, for receiving the output channel video signalsand a channel selector, e.g., 1171-CS, which transmits connectionrequests via a communication path 1181 to a network controller 1180included in network 1100.

Network 1100 includes 12, 8×1 first stage switches 1110-0 through1110-11, each having eight inlets and one outlet, and a single 12×6second stage switch 1190 having each of 12 inlets connected to anassociated one of the first stage switches 1110-0 through 1110-11 andhaving each of six outlets connected to one of the output channels OC0through OC5. The 32 input channels are connected to the 96 first stageswitch inlets by a connection arrangement 1140. Connection arrangement1140 connects each first stage switch inlet to an associatedpredetermined one of the input channels IC0 through IC31 in accordancewith the 3×16 connection matrix of FIG. 9. Each of the 16 columns of theconnection matrix is associated with two of the input channels IC0through IC31. For example, column 0 is associated with input channelsIC0 and IC16, column 1 is associated with input channels ICl and IC17,column 2 is associated with input channels IC2 and IC18, etc. Each firststage switch has a unique switch designation 0, 1, 2 . . . 11. A givencolumn of the connection matrix defines that the input channelsassociated with the given column are connected to the first stageswitches having designations in the given column. Column 0 defines thatinput channels IC0 and IC16 are connected to first stage switches1110-0, 1110-4 and 1110-8 having designations 0, 4 and 8 respectively.Column 1 defines that input channels ICl and IC17 are connected to firststage switches 1110-0, 1110-5 and 1110-9 having designations 0, 5 and 9,respectively. Column 2 defines that input channels IC2 and IC18 areconnected to first stage switches 1110-0, 1110-6 and 1110-10 havingdesignations 0, 6 and 10, respectively, etc. Several observations can bemade concerning connection arrangement 1140. First, each of the inputchannels IC0 through IC31 is connected to exactly three first stageswitch inlets. Input channel IC0, for example, is connected to inlets offirst stage switches 1110-0, 1110-4 and 1110-8. Second, no pair of firststage switches intersect in more than two input channels. An importantcharacteristic of connection arrangement 1140 can be stated as follows.For any group of six of the input channels IC0 through IC31, there is agroup of six of the first stage switches 1110-0 through 1110-11, eachhaving one inlet connected to a different one of that group of inputchannels. For example, consider the group of input channels IC0, IC4,IC8, IC12, IC16 and IC18. Each switch of the group of first stageswitches 1110-0, 1110-1, 1110-2, 1110-3, 1110-4 and 1110-6 has one inletconnected to a different one of that group of input channels. Switch1110-0 has an inlet connected to input channel IC0, switch 1110-1 has aninlet connected to input channel IC4, switch 1110-2 has an inletconnected to input channel IC8, switch 1110-3 has an inlet connected toinput channel IC12, switch 1110-4 has an inlet connected to inputchannel IC16 and switch 1110-6 has an inlet connected to input channelIC18. It is possible that for certain sequences of the customerfacilities 1171 through 1176 transmitting connection requests for inputchannels IC0, IC4, IC8, IC12 and IC16, network 1100 may temporarilyblock one or more of the requested input channels. However, it is alwayspossible to rearrange the connections of the first stage switches suchthat switches 1110-0, 1110-1, 1110-2, 1110-3, 1110-4 and 1110-6 connectinput channels IC0, IC4, IC8, IC12, IC16, and IC18, respectively, toinlets of second stage switch 1190. The connections within the secondstage switch 1190 can then be rearranged such that the input channelsIC0, IC4, IC8, IC12, IC16 and IC18 are connected to the customerfacilities 1171 through 1176 in accordance with the connection requests.Since this is possible for any group of six of the input channels IC0through IC31, network 1100 is a rearrangeable broadcast network. Thearrangement of connections within the 12 first stage switches 1110-0through 1110-11 and the second stage switch 1190 is controlled bynetwork controller 1180 via two communication paths 1182 and 1183.

FIG. 10 is a block diagram of a two-stage, rearrangeable broadcastnetwork 1200 that is equivalent to network 1100. The 12×6 second stageswitch 1290 of network 1200 is identical to second stage switch 1190 ofnetwork 1100. However the connection arrangement 1140 and first stageswitches 1110-0 through 1110-5 of network 1100 are replaced with apartial concentrator 1250 in network 1200. Partial concentrator 1250comprises 32 columns connected to the input channels IC0 through IC31,and 12 rows connected to the 12 inlets of second stage switch 1290.Partial concentrator 1250 includes two copies of a 16-column partialconcentrator. The 12 rows comprise three sets of four rows. For anyinteger i, 0≦i≦3, and any integer j, 0≦j≦2, row i of set j is incidentto columns corresponding to columns of the orthogonal array of FIG. 8,which in row j of the array, contain symbols defining row i. Forexample, row 0 of set 0 is incident to columns 0, 1, 2 and 3 sincecolumns 0, 1, 2 and 3 of the orthogonal array of FIG. 8 have the symbol0 in row 0. Similarly, row 3 of set 1 is incident to columns 3, 7, 11and 15 since columns 3, 7, 11 and 15 of the orthogonal array have thesymbol 3 in row 1. Note that the crosspoint pattern of columns 16through 31 of partial concentrator 1250 is identical to that for columns0 through 15.

A partial concentrator is called resolvable if its rows can bepartitioned into M groups such that each column is incident to exactlyone row in each group. Partial concentrator 1250 is resolvable since itsrows can be partitioned into three groups such that each column isincident to exactly one row in each group.

FIG. 11 is a block diagram of a two-stage rearrangeable broadcastnetwork 1300 constructed by extending network 1100 previously described.Network 1300 is used to interconnect 32 input channels OC0 through OC31to 18 output channels OC0 through OC17. Connection arrangement 1340 ofnetwork 1300 is identical to connection arrangement 1140 of network1100. Twelve first stage switches 1310-0 through 1310-11 are each 8×3rectangular switches in contrast to the twelve 8×1 first stage switches1110-0 through 1110-11 of network 1100. Network 1300 also includes three12×6 rectangular second stage switches 1390-0, 1390-1 and 1390-2 eachidentical to the one second stage switch 1190 of network 1100.

FIG. 12 is a block diagram of a two-stage rearrangeable broadcastnetwork 1400 representing another extension of the network 1100construction Network 1400 is used to interconnect 32 input channels IC0through IC31 to n₂ output channels OC0 through OC(n₂ -1). Connectionarrangement 1440 of network 1400 is identical to connection arrangement1140 of network 1100 and connection arrangement 1340 of network 1300.Twelve 8×3 first stage switches 1410 through 1410-11 are identical tofirst stage switches 1310-0 through 1310-11 of network 1300. However, innetwork 1400, all three outlets of each first stage switch are connectedto a 36×n₂ rectangular second stage switch 1490. By having multiplelinks from each first stage switch to second stage switch 1490, network1400 can serve greater than six output channels and still berearrangeable to avoid blocking. More generally, each first stage switchcould have x links to the second stage switch.

FIG. 13 is a block diagram of a generalized, two-stage broadcast network1500 constructed from an orthogonal array of order N and depth M.Network 1500 is used to interconnect N₁ input channels IC0 through IC(N₁-1) to n₂ output channels OC0 through OC(n₂ -1). Network 1500 includesv, r×1 first stage switches, e.g., 1501, 1502, a v×n₂ second stageswitch 1590, and a connection arrangement 1540. Connection arrangement1540 connects each of the inlets of the first stage switches to anassociated predetermined one of the input channels in accordance with aM×N² connection matrix. The first stage switches have unique switchdesignations d₀, d₁, . . . d_(v-1). For any integer i, 0≦i≦M-1, theelements of row i of the connection matrix are obtained by adding iN tocorresponding elements of an orthogonal array of order N and depth M Theorthogonal array has symbols 0,1, . . . N-1. Each column of the matrixis associated with w of the input channels Each element e of a givencolumn of the matrix defines that the input channels associated with thegiven column are connected to the first stage switch having the switchdesignation d_(e). The construction parameters for network 1500 aresummarized in Table 2.

                  TABLE 2                                                         ______________________________________                                        A M × N.sup.2 connection matrix corresponding to                        an orthogonal array of order N and depth M is                                 used to construct connection arrangement                                      1540. Each column of the connection matrix                                    is associated with w input channels.                                          N.sub.1 > 1, n.sub.2 > 1, n.sub.2 ≦ N.sub.1, v ≧ n.sub.2, r     > 1,                                                                          M > 1, N > 1, w > 1, wN.sup.2 ≧ N.sub.1, MN ≧ v,                wN = r                                                                        CONSTRUCTION PARAMETERS                                                       FOR NETWORK 1500                                                              ______________________________________                                    

Networks can be derived from network 1500 for serving fewer inputchannels by deleting columns of the connection matrix and, if possible,eliminating first stage switches. Each of the input channels IC0 throughIC(N₁ -1) is connected to exactly M first stage switch inlets. No pairof first stage switches intersect in more than w input channels (thew-intersect property). For any group of n₂ of the input channels IC0through IC(N₁ -1), there is a group of n₂ of the first stage switches,e.g., 1501, 1502, each having one inlet connected to a different one ofthat group of input channels. It has been determined by mathematicalanalysis that network 1500 is assured to be a rearrangeable, broadcastnetwork by having n₂ at most equal to

    (M-1) (M/w+1)+M/w                                          (3)

where y denotes the smallest integer not less than y and z denotes thelargest integer not exceeding z. Of course network 1500 can be extendedto serve a greater number, n₂, of output channels by having multiplelinks between each first stage switch and the second stage switch as innetwork 1400 (FIG. 12). If there are x links between each first stageswitch and the second stage switch, the network is a rearrangeable,broadcast network where n₂ at most equal to ##STR1## Network 1500 canalso be extended by having multiple second stage switches as in network1300 (FIG. 11).

FIG. 24 is a block diagram of a two-stage, rearrangeable broadcastnetwork 6300 that is equivalent to network 1300 (FIG. 11). The 12×6rectangular switches 6390-0, 6390-1, and 6390-2 are identical to thesecond stage switches 1390-0, 1390-1, and 1390-2 of network 1300.However the connection arrangement 1340 and first stage switches 1310-0through 1310-11 of network 1300 are replaced with three identicalpartial concentrators 6350-0, 6350-1, and 6350-2 in network 6300. Eachof the partial concentrators 6350-0, 6350-1, and 6350-2 is identical topartial concentrator 1250 of network 1200 (FIG. 10). Of course the threepartial concentrators 6350-0, 6350-1, and 6350-2 can also be representedas a single larger partial concentrator. Further the rows of the singlelarger partial concentrator can be reordered arbitrarily for practicaldesign considerations, for example such that identical rows areconsecutive.

FIG. 25 is a block diagram of a two-stage, rearrangeable broadcastnetwork 6400 that is equivalent to network 1400 (FIG. 12). The 36×n₂rectangular switch 6490 is identical to second stage switch 1490 ofnetwork 1400. However the connection arrangement 1440 and first stageswitches 1410-0 through 1410-11 of network 1400 are replaced with three(or, more generally, x) identical partial concentrators 6450-0, 6450-1,and 6450-2 in network 6400. Each of the partial concentrators 6450-0,6450-1, and 6450-2 is identical to partial concentrator 1250 of network1200 (FIG. 10).

NETWORK CONSTRUCTIONS BASED ON DIFFERENCE SETS

FIG. 14 is a block diagram of a two-stage, rearrangeable broadcastnetwork 2100 constructed based on a combinatorial design referred to asa difference set. FIG. 15 shows two particular difference sets D₁ and D₂used to construct network 2100. A set of r residues D:{a₁, . . . ,a_(r)} modulo u is called a (u,r,w) difference set if for every d≢0 (modulou), there are exactly w ordered pairs (a_(i),a_(j)), where a_(i) anda_(j) are elements of D such that a_(i) -a_(j) ≡d (modulo u). Adifference table for the (7,4,2) difference set D₁ is shown in FIG. 16.The rows and columns of the difference table are labelled with theelements 0, 2, 3 and 4 of D₁. Each cell entry of the difference table isthe difference resulting from the modulo 7 subtraction of the columnheading from the row heading. Note that in the difference table of FIG.16, every number from 1 to 6 appears exactly twice. In general, the cellentries in the difference table for a (u,r,w) difference set arecalculated modulo u, every number from 1 to u-1 appears exactly w timesin the difference table, and the difference set has r elements.Difference sets are well known combinatorial structures. Themathematical properties of difference sets and the techniques forconstructing such designs are described, for example, in Chapter 11 ofthe above-referenced book by Hall, and the publication of Leonard D.Baumert, Cyclic Difference Sets, (1971), Springer-Verlag.

Network 2100 is constructed from the two (7,4,2) difference sets D₁ andD₂ shown in FIG. 15. D₂ is obtained from D₁ by multiplying modulo 7 eachof the elements of D₁ by two. The 21×4 connection matrix shown in FIG.17 is used as an intermediate step in the construction of network 2100.The matrix of FIG. 17 is obtained from the difference sets D₁ and D₂ asfollows. The 21 rows of the matrix comprise three sets, s₀, s₁ and s₂,each having seven rows. Each of the numbers 0, 1,. . . , 27 is placedarbitrarily in the set s₀. The columns of s₁ are determined from D₁={0,2,3,4} as follows. Column 0 of s₁ is the same as column 0 of s₀.Column 1 of s₁ is obtained by rotating column 1 of s₀ by 2 positions.Column 2 of s₁ is obtained by rotating column 2 of s₀ by 3 positions.Column 3 of s₁ is obtained by rotating column 3 of s₀ by 4 positions.The columns of s₂ are determined from D₂ ={0,4,6,1} as follows. Column 0of s₂ is the same as column 0 of s₀. Column 1 of s₂ is obtained byrotating column 1 of s₀ by 4 positions. Column 2 of s₂ is obtained byrotating column 2 of s₀ by 6 positions. Column 3 of s₂ is obtained byrotating column 3 of s₀ by 1 position.

Network 2100 (FIG. 14) is used to broadcast video signals from 28 videovendors, e.g., 2151 and 2152, to 13 customer facilities 2161 through2173. Network 2100 receives video signals in 28 input channels IC0through IC27 and transmits video signals in 13 output channels OC0through OC12 each connected to one of the customer facilities 2161through 2173. Each of the customer facilities 2161 through 2173 includesa receiver, e.g., 2161-R, for receiving the output channel video signalsand a channel selector, e.g., 2161-CS, which transmits connectionrequests via a communication path 2181 to a network controller 2180included in network 2100.

Network 2100 includes 21, 4×1 first stage switches 2110-0 through2110-20, each having four inlets and one outlet, and a single 21×13second stage switch 2190 having each of 21 inlets connected to anassociated one of the first stage switches 2110-0 through 2110-20 andhaving each of 13 outlets connected to one of the output channels OC0through OC12. The 28 input channels are connected to the 84 first stageswitch inlets by a connection arrangement 2140. Connection arrangement2140 connects each first stage switch inlet to an associatedpredetermined one of the input channels IC0 through IC27 in accordancewith the 21×4 connection matrix of FIG. 17. The elements of the matrixare the channel designations 0,1, . . . ,27 of the input channels IC0through IC27. Each of the 21 rows of the matrix is associated with oneof the 21 first stage switches. The channel designations occurring in agiven row of the matrix define the ones of the input channels connectedto the inlets of the first stage switch associated with the given row.For example, row 0 defines that input channels IC0, IC7, IC14, and IC21are connected to first stage switch 2110-0. Row 1 defines that inputchannels ICl, IC8, IC15, and IC22 are connected to first stage switch2110-1. Row 2 defines that input channels IC2, IC9, IC16, and IC23 areconnected to first stage switch 2110-2, etc. Several observations can bemade concerning connection arrangement 2140. First, each of the inputchannels IC0 through IC27 is connected to exactly three first stageswitch inlets. Input channel IC0, for example, is connected to inlets offirst stage switches 2110-0, 2110-7 and 2110-14. Second, considering thefirst stage switches in three groups 2110-0 through 2110-6, 2110-7through 2110-13, and 2110-14 through 2110-20, for any two first stageswitches in a given group there are at most two first stage switches ineach other group that intersect both of the two first stage switches inthe given group. An important characteristic of connection arrangement2140 can be stated as follows. For any group of 13 of the input channelsIC0 through IC27, there is a group of 13 of the first stage switches2110-0 through 2110-20, each having one inlet connected to a differentone of that group of input channels. For example, consider the group of13 input channels IC0 through IC12. Each switch of the group of 13 firststage switches 2110-0 through 2110-6, 2110-9 through 2110-13, and2110-16 has one inlet connected to a different one of that group ofinput channels. It is possible that for certain sequences of thecustomer facilities 2161 through 2173 transmitting connection requestsfor input channels IC0 through IC12, network 2100 may temporarily blockone or more of the requested input channels. However, it is alwayspossible to rearrange the connections of the first stage switches suchthat switches 2110-0 through 2110-6, 2110-9 through 2110-13, and 2110-16connect input channel IC0 through IC12 to inlets of second stage switch2190. The connections within the second stage switch 2190 can then berearranged such that the input channels IC0 through IC12 are connectedto the customer facilities 2161 through 2173 in accordance with theconnection requests. Since this is possible for any group of 13 of theinput channels IC0 through IC27, network 2100 is a rearrangeablebroadcast network. The arrangement of connections within the 21 firststage switches 2110-0 through 2110-20 and the second stage switch 2190is controlled by network controller 2180 via two communication paths2182 and 2183.

FIG. 18 is a block diagram of a two-stage, rearrangeable broadcastnetwork that is equivalent to network 2100. The 21×13 second stageswitch 2290 of network 2200 is identical to second stage switch 2190 ofnetwork 2100. However the connection arrangement 2140 and first stageswitches 2110-0 through 2110-20 of network 2100 are replaced with apartial concentrator 2250 in network 2200. Partial concentrator 2250comprises 28 columns connected to the input channels IC0 through IC27,and 21 rows connected to the 21 inlets of second stage switch 2290. Inpartial concentrator 2250, each of the 21 rows is incident to ones ofthe 28 columns defined by a corresponding row of the 21×4 connectionmatrix of FIG. 17. For example, row 0 is incident to columns 0, 7, 14and 21. Row 1 is incident to columns 1, 8, 15 and 22. Row 2 is incidentto columns 2, 9, 16 and 23 etc. The 21 rows of partial concentrator 2250comprise three sets of seven rows each. Note that for any two rows in agiven set, there exists at most two rows in each other set whichintersect both of the two rows in the given set.

Partial concentrator 2250 is resolvable since its rows can bepartitioned into three groups such that each column is incident toexactly one row in each group. A resolvable concentrator is said to havethe w-double-intersect property if for any two rows in a given groupthere exist at most w rows in each other group that intersect both ofthe two rows in the given group. Partial concentrator 2250 has thew-double-intersect property with w equal to two.

FIG. 19 is a block diagram of a two-stage rearrangeable broadcastnetwork 2300 constructed by extending network 2100 previously described.Network 2300 is used to interconnect 28 input channels IC0 through IC27to 26 output channels OC0 through OC25. Connection arrangement 2340 ofnetwork 2300 is identical to connection arrangement 2140 of network2100. Twenty-one first stage switches 2310-0 through 2310-20 are each4×2 rectangular switches in contrast to the 21, 4×1 first stage switchesof network 2100. Network 2300 also includes two 21×13 rectangular secondstage switches 2390-0 and 2390-1 each identical to the one second stageswitch 2190 of network 2100.

FIG. 20 is a block diagram of a two-stage rearrangeable broadcastnetwork 2400 representing another extension of the network 2100construction. Network 2400 is used to interconnect 28 input channels IC0through IC27 to n₂ output channels OC0 through OC(n₂ -1). Connectionarrangement 2440 of network 2400 is identical to connection arrangement2140 of network 2100 and connection arrangement 2340 of network 2300.Twenty-one 4×2 first stage switches 2410-0 through 2410-20 are identicalto first stage switches 2310-0 through 2310-20 of network 2300. However,in network 2400, both outlets of each first stage switch are connectedto a 42×n₂ rectangular second stage switch 2490. By having multiplelinks from each first stage switch to second stage switch 2490, network2400 can serve greater than 13 output channels and still berearrangeable to avoid blocking. More generally, each first stage switchcould have x links to the second stage switch.

FIG. 21 is a block diagram of a generalized, two-stage broadcast network2500 constructed from M-1 (u,r,w) difference sets. Network 2500 is usedto interconnect N₁ input channels IC0 through IC(N₁ -1) to n₂ outputchannels OC0 through OC(n₂ -1). Network 2500 includes v, r×1 first stageswitches, e.g., 2501, 2502, a v×n₂ second stage switch 2590, and aconnection arrangement 2540. Connection arrangement 2540 connects eachof the inlets of the first stage switches to an associated predeterminedone of the input channels in accordance with a Mu×r connection matrix.Each of the input channels has a unique channel designation. Theconnection matrix is derived from M-1 (u,r,w) difference sets D₁, D₂, .. . , D_(M-1). First, a (u,r,w) difference set D₁ is constructed. Eachof the difference sets other than D₁ is derived by multiplying modulo uthe elements of D₁ by a residue modulo u that is relatively prime to u.The Mu rows of the connection matrix comprise M sets, s₀, s₁ , . . .s_(M-1), each having u rows. Each of the channel designations of theinput channels occurs exactly once in set s₀. For any integer i,1≦i≦M-1, for any integer j, 0≦j≦r-1, and for the M-1 (u,r,w) differencesets D_(i) ={a_(o), . . . a_(r-1) ^(i) }, column j of set si is obtainedby rotating column j of set s₀ by a_(j) ^(i) positions. (The superscripti does not denote exponentiation but rather identifies the elements suchas a_(o) ^(i) as belonging to the difference set D_(i).) The channeldesignations occurring in a given row of the connection matrix definethe ones of the input channels connected to the inlets of a given firststage switch associated with the given row. The construction parametersfor network 2500 are summarized in Table 3.

                  TABLE 3                                                         ______________________________________                                        A Mu × r connection matrix derived from M-1                             (u,r,w) difference sets is used to construct                                  connection arrangement 2540.                                                  N.sub.1 > 1, n.sub.2 > 1, n.sub.2 ≦ N.sub.1, v ≧ n.sub.2, r     > 1,                                                                          w > 0, M > 1, u > 1, ur ≧ N.sub.1, Mr ≧ v                       CONSTRUCTION PARAMETERS                                                       FOR NETWORK 2500                                                              ______________________________________                                    

Note that Mu is at least equal to v, the number of first stage switches,and ur is at least equal to N₁, the number of input channels. Networkscan be derived from network 2500 for serving fewer input channels bydeleting cells from the connection matrix and, if possible, eliminatingfirst stage switches. Each of the input channels IC0 through IC(N₁ -1)is connected to exactly M first stage switch inlets. Considering thefirst stage switches in M groups, for any two first stage switches in agiven group there are at most w first stage switches in each other groupthat intersect both of the two first stage switches in the given group(the w-double-intersect property). For any group of n₂ of the inputchannels IC0 through IC(N₁ -1), there is a group of n₂ of the firststage switches, e.g., 2501, 2502, each having one inlet connected to adifferent one of that group of input channels. It has been determined bymathematical analysis that network 2500 is a rearrangeable, broadcastnetwork where n₂ is at most equal to

    M.sup.3 -M.sup.2)/w+M+1                                    (5)

where z denotes the largest integer not exceeding z. course network 2500can be extended to serve a greater number, n₂, of output channels byhaving multiple links between each first stage switch and the secondstage switch as in network 2400 (FIG. 20). If there are x links betweeneach first stage switch and the second stage switch, the network is arearrangeable broadcast network where n₂ is at most equal to

    x.sup.3 M.sup.3 -x.sup.2 M.sup.2)/w+xM+1,                  (6)

Network 2500 can also be extended by having multiple second stageswitches as in network 2300 (FIG. 19).

FIG. 26 is a block diagram of a two-stage, rearrangeable broadcastnetwork 7300 that is equivalent to network 2300 (FIG. 19). The 21×13rectangular switches 7390-0 and 7390-1 are identical to the second stageswitches 2390-0 and 2390-1 of network 2300. However the connectionarrangement 2340 and first stage switches 2310-0 through 2310-20 ofnetwork 2300 are replaced with two identical partial concentrators7350-0 and 7350-1 in network 7300. Each of the partial concentrators7350-0 and 7350-1 is identical to partial concentrator 2250 of network2200 (FIG. 18). Of course the two partial concentrators 7350-0 and7350-1 can also be represented as a single larger partial concentrator.Further the rows of the single larger partial concentrator can bereordered arbitrarily for practical design considerations, for examplesuch that identical rows are consecutive.

FIG. 27 is a block diagram of a two-stage, rearrangeable broadcastnetwork 7400 that is equivalent to network 2400 (FIG. 20). The 42×n₂rectangular switch 7490 is identical to second stage switch 2490 ofnetwork 2400. However the connection arrangement 2440 and first stageswitches 2410-0 through 2410-20 of network 2400 are replaced with two(or, more generally, x) identical partial concentrators 7450-0 and7450-1 in network 7400. Each of the partial concentrators 7450-0 and7450-1 is identical to partial concentrator 2250 of network 2200 (FIG.18).

What is claimed is:
 1. A switching network for selectively connecting N₁input channels to n₂ output channels such that none of said n₂ outputchannels is connected to more than one of said N₁ input channels at anytime, N₁ and n₂ being positive integers each greater than one, n₂ beingat most equal to N₁, said switching network comprisinga partialconcentrator having N₁ columns each connected to a different one of saidN₁ input channels, and v rows, said partial concentrator comprisingmeans for selectively connecting ones of said columns to ones of saidrows, v being a positive integer at least equal to n₂, and a switchhaving n₂ outlets each connected to a different one of said outputchannels and v inlets each connected to a different one of said rows,said switch comprising means for selectively connecting said outlets tosaid inlets, said partial concenttrator being derivable from a partialconcentrator having c columns and b rows and having each of said ccolumns incident to ones of said b rows defined by a corresponding rowof a (b,c,r,M,w) block design, where a column is said to be incident toa row if there is a crosspoint at the intersection, where each of said ccolumns has a unique column designation, where the elements of saidblock design are the column designations of said c columns, where b,c,r,and M are positive integers each greater than one, where w is a positiveinteger, where M+w is greater than three, where b is at least equal tov, and where c is at least equal to N₁.
 2. A switching network inaccordance with claim 1 wherein n₂ is at most equal to (M² -1)/w+1.
 3. Aswitching network for selectively connecting N₁ input channels to n₂output channels such that none of said n₂ output channels is connectedto more than one of said N₁ input channels at any time, N₁ and n₂ beingpositive integers each greater than one, n₂ being at most equal to N₁,said switching network comprisinga plurality of x partial concentratorseach having N₁ columns each connected to a different one of said N₁input channels, and v rows, said each partial concentrator comprisingmeans for selectively connecting ones of said columns to ones of saidrows, x being a positive integer greater than one and v being a positiveinteger at least equal to n₂, and a switch having n₂ outlets eachconnected to a different one of said output channels and xv inlets eachconnected to a different one of said rows of said x partialconcentrators, said switch comprising means for selectively connectingsaid outlets to said inlets, said each partial concentrator beingderivable from a partial concentrator having columns and b rows andhaving each of said c columns incident to ones of said b rows defined bya corresponding row of a (b,c,r,M,w) block design, where a column issaid to be incident to a row if there is a crosspoint at theintersection, where each of said c columns has a unique columndesignation, where the elements of said block design are the columndesignations of said c columns, where b,c,r and M are positive integerseach greater than one, where w is a positive integer, where M+w isgreater than three, where b is at least equal to v, and where c is atleast equal to N₁.
 4. A switching network in accordance with claim 3wherein n₂ is at most equal to x[(xM+1) (M-1)/w+1].
 5. A switchingnetwork for selectively connecting N₁ input channels to n₂ outputchannels such that none of said n₂ output channels is connected to morethan one of said N₁ input channels at any time, N₁ and n₂ being positiveintegers each greater than one, n₂ being at most equal to N₁, saidswitching network comprisinga plurality of x partial concentrators eachhaving N₁ columns each connected to a different one of said N₁ inputchannels, and v rows, said each partial concentrator comprising meansfor selectively connecting ones of said columns to ones of said rows, xbeing a positive integers greater than one and v being a positiveinteger at least equal to n₂, and a switch having n₂ outlets eachconnected to a different one of said output channels and xv inlets eachconnected to a different one of said rows of said x partialconcentrators, said switch comprising means for selectively connectingsaid outlets to said inlets, said each partial concentrator beingderivable from a partial concentrator having c columns and b rows andhaving eahc of said c columns incident to ones of said b rows defined bya corresponding row of a (b,c,r,M,1) block design, where a column issaid to be incident to a row if there is a crosspoint at theintersection, where eahc of said c columns has a unique columndesignation, where the elements of said block design are the columndesignations of said c columns, where b,c,r, and M are positive integerseach greater than one, where b is at least equal to v, and where c is atleast equal to N₁.
 6. A switching network for selectively connecting N₁input channels to N₂ output channels such that none of said N₂ outputchannels is connected to more than one of said N₁ input channels at anytime, N₁ and N₂ being positive integers each greater than one, saidswitching network comprisinga plurality of partial concentrators eachhaving N₁ columns each connected to a different one of said n₁ inputchannels, and v rows, said each partial concentrator comprising meansfor selectively connecting ones of said columns to ones of said rows, vbeing a positive integer at least equal to a positive integer, n₂, thatis less than N₂ and at most equal to N₁, and a plurality of switcheseach associated with a different one of said partial concentrators andeach having n₂ outlets each connected to a different one of said outputchannels and v inlets each connected to a different one of the v rows ofthe associated partial concentrator, said each switch comprising meansfor selectively connecting said outlets of said each switch to saidinlets of said each switch, said each partial concentrator beingderivable from a partial concentrator having c columns and b rows andhaving each of said c columns incident to ones of said b rows defined bya corresponding row of a (b,c,r,M,w) block design, where a column issaid to be incident to a row if there is a crosspoint at theintersection, where each of said c columns has a unique columndesignation, where the elements of said block design are the columndesignations of said c columns, where b,c,r, and M are positive integerseach greater than one, where w is a positive integer, where M+w isgreater than three, where b is at least equal to v, and where c is atleast equal to N₁.